curve #3636

y² + xy + y = x³ + x² − 817732582601069158567499815x + 9294691321682809931409995675785428600797

a-invariants: [1, 1, 1, -817732582601069158567499815, 9294691321682809931409995675785428600797]
rank (lower bound): ≥ 13
torsion subgroup: ℤ/2ℤ
conductor (N): 142345695678637990158711155259549347070
discriminant (Δ): -2325400601197026866769054026736995379255314916994583455181126401087272284160000000
Faltings height: 14.3824
naive height: 197.5837
primes of bad reduction: 2, 3, 5, 7, 11, 17, 19, 41, 43, 61, 71, 89, 101, 107, 241, 439, 607, 2203, 2447, 75037
regulator: 34948568724930.17615654503194164794372280016036127466468845097663
submitted by: Michael Rubinstein
submitted at: 2026-09-29 19:24:06 UTC
last updated: 2026-09-29 19:24:06 UTC

Witness: 13 independent points log in to add more points →

(-32125316545493, 49093064295994145946)
(-30680015895133, 74194116435422433906)
(-20131942379573, 132656931772193654106)
(16993362554837, 17490350397031078386)
(18259099096907, 21239527677838304346)
(107582018835707, 1080029531343693169146)
(333358188622243/9, 4662694715294001126782/27)
(20322673562771603/1369, 1046592310395283200252978/50653)
(21833417777310437/841, 1819045154717885721773994/24389)
(1200179143607378467/84681, 578941653401145215323435966/24642171)
(3389457379306819627/40401, 5918480260584382539640401466/8120601)
(3903463771915938323/458329, 16847399168896210707668929458/310288733)
(4884479909441928667/269361, 2895810411952166941768109974/139798359)

Commentary

Specialization at T = 311/126 of the rank-9 fibration of X400, y^2 + (Lx+B)y = x(x+D)(x+E), given in the commentary of curve #802 (same T), found by searching small-height t = a/b for small conductor, with points from a 2-descent (PARI ellrank).

last edited by Michael Rubinstein at 2026-09-29 19:24:06 UTC · history